For the \(n\) -vertex path \(P_n\) , Erdős [1]} asked if \(\hat{r}_2(P_n)=O(n)\) , which was
answered positively by Beck [2]} using the probabilistic method.
An explicit construction for the same results was given by Alon and
Chung [3]}.
Many successive improvements led to the currently best known bounds \(3.75n -o(n)\le \hat{r}_2(P_n)\le 74n\)  (see, e.g.,
[2]}, [5]}, [6]}, [7]} for lower
bounds, and [2]}, [9]}, [10]}, [6]} for
upper bounds).
For \(s\ge 2\)  colours, Dudek and Prałat [6]} and
Krivelevich [13]} proved that there are constants \(c\)  and
\(C\)  such that \(cs^2n\le \hat{r}_s(P_n)\le Cs^2(\log s)n\) .
